Locating and Computing All the Simple Roots and Extrema of a Function

Abstract

This paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function f: (a, b) C N --R and all the extrema of a three times continuously differentiable function in (a, b). The first algorithm locates and computes all the simple roots or all the extrema, while the second one is more efficient in the case where both simple roots and extrema are required.This paper also gives analytical estimation of the expected complexity of the algorithms based on the distribution of the roots in (a, b). Here only the case of uniform distribution is examined, which is also the approach to be followed when no statistical data are available for the function at hand.The algorithms have been implemented and tested. Performance information for a well-known Bessel function is reported.

On perturbation of roots of homogeneous algebraic systems

“…In particular, our Theorem 5.4 assures the existence of a deformed system (5.4) of the original system (2.1) that possesses only simple real roots. This result can be used in many cases including the computation of the topological degree [4,6,10,11,12] in order to examine the solution set of a system of equations and to obtain information on the existence of solutions, their number and their nature [1,3,6,8,9].…”

Global Optimization for Imprecise Problems

“…One way to estimate this number is the usage of degree computational techniques [25]. For this purpose one can apply Picard's theorem and compute the value of the topological degree of the extended Picard's function [24]. For the computation of this value Aberth's method [1], which is an adaptation of Kearfott's method [28], can be utilized.…”

“…Or, in other cases, it may be necessary to integrate numerically a system of differential equations in order to obtain a function value, so that the precision of the computed value is limited [29,48]. On the other hand, it is necessary, in many applications, to use methods which do not require precise values [48,24], as for example in neural network training [31,32,30]. Furthermore, in many problems the values of the function to be minimized are computationally expensive [23].…”

On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

“…The importance of the problem has attracted the attention of many research efforts and, as a result, many different approaches to the problem exist. We briefly mention here the deflation techniques used for the calculation of further solutions [7] or other more efficient and more recent interval analysis based methods (see, e.g., [15,16,26,28,30,37]) and the methods described in [20,21,41]. The corresponding existence tool of interval analysis based methods is the availability of the range of the function in a given interval, which can be implemented using interval arithmetic, though range overestimation, and hence efficiency problems must be resolved.…”

“…Assume further that f(a) f(b) ] 0 and that all the roots of f that lie in (a, b) are simple. Then, by applying (3.1) for n=2 we obtain that the total number N r of roots of f that lie in (a, b) is given by [21],…”